Identifying the invariants for classical knots and links from the YokonumaHecke algebras
Abstract
In this paper we announce the existence of a family of new $2$variable polynomial invariants for oriented classical links defined via a Markov trace on the YokonumaHecke algebra of type $A$. YokonumaHecke algebras are generalizations of IwahoriHecke algebras, and this family contains the Homflypt polynomial, the famous $2$variable invariant for classical links arising from the IwahoriHecke algebra of type $A$. We show that these invariants are topologically equivalent to the Homflypt polynomial on knots, but not on links, by providing pairs of Homflyptequivalent links that are distinguished by our invariants. In order to do this, we prove that our invariants can be defined diagrammatically via a special skein relation involving only crossings between different components. We further generalize this family of invariants to a new $3$variable skein link invariant which is stronger than the Homflypt polynomial. Finally, we present a closed formula for this invariant, by W.B.R. Lickorish, which uses Homflypt polynomials of sublinks and linking numbers of a given oriented link.
 Publication:

arXiv eprints
 Pub Date:
 May 2015
 arXiv:
 arXiv:1505.06666
 Bibcode:
 2015arXiv150506666C
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Representation Theory;
 57M27;
 57M25;
 20F36;
 20F38;
 20C08
 EPrint:
 57 pages, 12 figures, 1 table. For related computational packages, see http://math.ntua.gr/~sofia/yokonuma . Added an appendix by W.B.R. Lickorish